1. On the ghost sector of OSFT Carlo Maccaferri SFT09, Moscow Collaborators: Loriano Bonora, Driba Tolla

2. Motivations We focus on the oscillator realization of the gh=0 star algebra Fermionic ghosts have a “clean” 3 strings vertex at gh=1,2 (Gross, Jevicki) We need a formulation on the SL(2,R)-invariant vacuum to be able to do (for example) is a squeezed state on the gh=0 vacuum, How do such squeezed states star-multiply? Is it possible to have in critical dimension?

3. Plan of the talk Surface states as squeezed states Surface states with ghost insertions as squeezed states K1 invariance The midpoint basis, decoupling of zero modes Star product of wedges via the N string vertex Reconstruction of gh=0 and gh=3 wedges from spectral data The norm of wedge states in the critical dimension Further perspectives

4. Surface States as squeezed states Given a map

5. This is a good representation because The squeezed form exactly captures all the n-point functions

6. Surfaces with insertions as squeezed states Surfaces with k c-insertions are also squeezed states on the gh=k vacuum With the neumann function given by Again 2n-point functions are given by the determinant of n 2-point function, so the squeezed state rep is consistent To reflect a surface to gh=3 we can use the BRST invariant insertion of

7. Invariance On the gh=0 vacuum we have On the gh=3 vacuum K1 invariance does not mean commuting nemann coefficients

8. The reason is in the vacuum doublet But Is it possible to have K1 invariance at gh=3? The obvious guess is given by But this is not a squeezed state (but a sum of two) (very different from the gh=1/gh=2 doublet, or to the h=(1,0) bc-system) Our aim is to define gh=3 “mirrors” for all wedge states, which are still squeezed states with non singular neumann coefficients (bounded eigenvalues) and which are still annihilated by K1

9. Reduced gh=3 wedges Consider the Neumann function for the states LT analysis shows diverging eigenvalues, indeed Real and bounded eigenvalues <1Rank 1 matrix (1 single diverging eigenvalue) We thus define reduced gh=3 wedges as Still we have

10. Midpoint Basis “Adapting” a trick by Okuyama (see also Gross-Erler) we can define a convenient gh=3 vacuum We need to redefine the oscillators on the new gh=0/gh=3 doublet by means of the unitary operator Reality Potentially dangerous Same as in gh=1/gh=2 We will see that this structure is also encoded in the eigenbasis of K1

11. On the vacua we have The oscillators are accordingly redefined Still we have And the fundamental

12. K1 in the midpoint basis Remember that K1 has the following form The midpoint basis just kills the spurious 3’s,

13. This very small simplification gives to squeezed states in the kernel of K1 the commuting properties that one would naively expect At gh=0 we have At gh=3 we have K1 invariance now implies commuting matrices

14. Gh=3 in the midpoint basis Going to the midpoint basis is very easy for gh=3 squeezed states The “bulk” part (non-zero modes) is unaffected The zero mode column mixes with the bulk for reduced gh=3 wedges For reduced states we thus have the non trivial identity

15. Gh=0 in the midpoint basis Here there are non normal ordered terms in the exponent, non linear relations In LT we also observe The midpoint basis is singular at gh=0, nontheless very useful as an intermediate step, because it effectively removes the difference between gh=0 and gh=3

16. The midpoint star product We want to define a vertex which implements For a N—strings vertex we choose the gluing functions (up to SL(2,R)) We start with the insertion of on the interacting worldsheet...It is a squeezed state but not a “surface” state (the surface would be the sum of 2 complex conjugated squeezed)...

17. Then we decompose Insertion functions Again, LT shows a diverging eigenvalue in the U’s

18. As for reduced gh=3 wedges we observe And therefore define Which very easily generalizes to N strings (3 N)

19. Properties Twist/bpz covariance K1 invariance Non linear identities (of Gross/Jevicki type) thanks to the “chiral” insertion

20. The vertex in the midpoint basis As for reduced gh=3 wedges, the vertex does not change in the bulk (non-zero modes) And it looses dependence on the zero modes So, even if zero modes are present at gh=0, they completly decouple in such a kind of product (isomorphism with the zero momentum matter sector) In particular, using the midpoint basis, it is trivial to show that

21. K1 spectroscopy K1 is well known to have a continuos spectrum, which manifests itself in continuous eigenvalues and eigenvectors of the matrices G and H Belov and Lovelace found the “bi-orthogonal” continuous eigenbasis of K1 for the bc system (our neumann coefficients are maps from the b-space to the c-space and vic.) Orthogonality “Almost” completeness RELATION WITH MIDPOINT BASIS

22. These are left/right eigenvectors of G However that’s not the whole spectrum of G The zero mode block has its own discrete spectrum

23. The discrete spectrum of G The zero mode matrix has eigenvalues Important to observe that

24. Normalizations Completeness relation

25. Spectroscopy in the midpoint basis Continuous spectrum with NO zero modes (both h=-1,2 vectors start from n=2) Discrete spectrum with JUST zero modes The midpoint basis confines the zero modes in the discrete spectrum (separate orthonormality for zero modes and bulk)

26. Reconstruction of BRST invariant states from the spectrum It turns out that all the points on the imaginary k axis are needed (not just ±2i) Wedge states eigenvalues have a pole in Given these poles, the wedge mapping functions are obtained from the genereting function of the continuous spectrum

27. Gh=3 Remembering the neumann function for Continuous spectrum Reduced gh=3 wedges Needed for BRST invariance

28. Gh=0 Zero modes Only for N=2 this coincides with the discrete spectrum of G (that’s the reason of the violation of commutativity) Once zero modes are (mysteriously) reconstructed, we can use the properties of the midpoint basis to get (and analytically compute)

29. The norm of wedge states As a check for the BRST consistency of our gh=0/gh=3 squeezed states, we consider the overlap (tensoring with the matter sector, so that c=0) Using Fuchs-Kroyter universal regularization (which is the correct way to do oscillator level truncation), we see that this is perfectly converging to 1 (for all wedges, identity and sliver included. n=3, m=3 n=3,m=30 n=1, m=7 n=1,m=1 Sliver Infinitely many rank 1 orthogonal projectors (RSZ, BMS) can be shown to have UNIT norm, see Ellwood talk, CP- factors (CM)